Four dimensional almost complex torus manifolds
Donghoon Jang
TL;DR
This work extends the toric correspondence to 4-dimensional almost complex torus manifolds by encoding fixed-point data into a pair of combinatorial gadgets: a finite family $\Delta$ of admissible multi-fans and a directed labeled graph $\Gamma$. It establishes necessary and sufficient conditions for these objects to describe a given manifold, and introduces a minimal model plus blow up/down operations that connect any description to a minimal one with unit-weight data. The results show that, in the complex case, $\Delta$ reduces to a fan determining $M$ and $\Gamma$ encodes the equivariant cohomology; crucially, any two 4D complex torus manifolds are connected by equivariant blow ups/downs and can be described by fans up to equivariant biholomorphism. The framework yields existence and classification results, including a construction from minimal data, a correspondence between graphs and multi-fans, and explicit connections to classical surfaces like $\mathbb{CP}^1\times\mathbb{CP}^1$ and Hirzebruch surfaces, with applications to Todd genus, $\chi_y$-genus, and Chern numbers. Overall, the paper provides a robust combinatorial-geometric toolkit for understanding 4D almost complex torus manifolds and their complex counterparts.
Abstract
In dimension 4, we extend the correspondence between compact nonsingular toric varieties and regular fans to a correspondence between almost complex torus manifolds and families of multi-fans in a geometric way, where an (almost) complex torus manifold is a $2n$-dimensional compact connected (almost) complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n$ that has fixed points. Let $M$ be a 4-dimensional almost complex torus manifold. To $M$, we associate two equivalent combinatorial objects, a family $Δ$ of multi-fans and a graph $Γ$, which encode the data on the fixed point set. We find a necessary and sufficient condition for each of $Δ$ and $Γ$. Moreover, we provide a minimal model and operations for each of $Δ$ and $Γ$. We introduce operations on a multi-fan and a graph that correspond to blow up and down of a manifold, and show that we can blow up and down $M$ to a minimal manifold $M'$ whose weights at the fixed points are unit vectors in $\mathbb{Z}^2$, $Δ$ to a family of minimal multi-fans that has unit vectors only, and $Γ$ to a minimal graph whose edges all have unit vectors as labels. As an application, if $M$ is complex, $Δ$ is a fan and determines $M$, $Γ$ encodes the equivariant cohomology of $M$, and $M'$ is $\mathbb{CP}^1 \times \mathbb{CP}^1$. This implies that any two 4-dimensional complex torus manifolds are obtained from each other by equivariant blow up and down.